About Me
I am a faculty member at the Department of Computer Science and Artificial Intelligence at Bar-Ilan University.
Prior to joining Bar-Ilan, I was a postdoc at the Computer Science Department at Columbia University, hosted by
Alex Andoni.
The postdoc was part of the Simons Collaboration on Algorithms and Geometry.
I did my PhD at the Computer Science Department at Ben-Gurion University of the Negev.
I was fortunate to be advised by Ofer Neiman and
Robert Krauthgamer.
My research stands at the exciting intersection of algorithms, geometry, and graph theory. Broadly speaking, my work focuses on summarization of geometric data. That is, ways to organize data of geometric nature in more manageable forms that could be utilized for algorithmic design. Some key words: low-distortion metric embeddings, stochastic decompositions, spanners, approximation algorithms, streaming algorithms, online algorithms, and computational geometry.
I am recruiting excellent graduate students to our research group. If you are interested, don't hesitate to shoot me an email!
Check out our Bar-Ilan Theory seminar !
Check out our FOCS 2022 workshop “ Advances on Metric Embeddings ” I co-organized with Hung Le. There are video recordings of all the talks!
Funding. I am grateful to the European Research Council (ERC) and the Israel Science Foundation (ISF) for funding my research.
Contact information
Arnold Filtser Email: arnold.filtser at biu dot ac dot ilOffice: Bar-Ilan University, bldg 503, room 307
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Video
This page includes several talks regarding my research papers.- How to Protect Yourself from Threatening Skeletons: Optimal Padded Decompositions for Minor-Free Graphs
[ Video , 65 min, Workshop on High-Dimensional and Complex Data Algorithms (Venice), May 2025]
- On Sparse partitions
[ Video , 45 min, DIMACS Workshop on Efficient Algorithms for High Dimensional Metrics: New Tools, May 2024]
- Locality-Sensitive Orderings
[ Video , 60 min, Bar-Ilan Theory Seminar, January 2023]
- Locality-Sensitive Orderings and Applications to Reliable Spanners
[ Video , 24 min, STOC 2022 presentation]
[ Video , 48 min, IDEAL Workshop on High-Dimensional Geometry and Analysis] - Survey: Metric Embeddings into Trees and its Various Spin-offs
[ Video , 83 min, Dagshtul 25212: Metric Sketching and Dynamic Algorithms for Geometric and Topological Graphs, May 2025]
- Metric Embedding into Trees
[ Video , 85 min, BIU lecture for undergrands, July 2024] Prospective students, start here!
[ Video , 64 min, BIU lecture for undergrands, May 2025] The most basic version of this lecture.
[ Video , 53 min, FOCS workshop on metric embeddings, November 2022]
[ Video , 83 min, HUJI theory seminar, March 2022] - Hop-Constrained Metric Embeddings and their Applications
[ Video , 20 min, FOCS 2021 presentation]
The matirial is also covered by the Metric Embedding into Trees talk. - Clan Embeddings into Trees, and Low Treewidth Graphs
[ Video , 29 min, STOC 2021 presentation]
[ Video , 67 min, Open University of Israel colloquium 2022, in Hebrew!]
The matirial is also covered by the Metric Embedding into Trees talk. - Graph Spanners by Sketching in Dynamic Streams and the Simultaneous Communication Model
[ Video , 25 min, SODA 2021 presentation] - Distributed Construction of Light Networks
[ Video , 20 min, PODC 2020 presentation] [ Slides ] - Scattering and Sparse Partitions, and their Applications
[ Video 1 , 45 min, recorded during Simons algorithms and geometry collaboration Monthly meeting, January 2020] [ Simons slides ]
[ Video 2, 60 min, recorded during Stony Brook (online) algorithms seminar, April 2020]
[ ICALP presentation, 23 min, ICALP 2020 presentation] [ ICALP slides ]
- On Strong Diameter Padded Decompositions
[ Video , 25 min] [ Slides ] - On metric embeddings, shortest path decompositions and face cover of planar graphs
[ Video , 45 min, recorded during a workshop on data science in low-dimensional spaces] - Metric Embedding via Shortest Path Decompositions
[ Video , 19 min, recorded during STOC 18] - Steiner Point Removal with distortion O(logk), using the Noisy-Voronoi algorithm
[ Video , 68 min] [ Slides ] - Ramsey Spanning Trees and their Applications
[ Video , 32 min] [ Slides ] - The Greedy Spanner is Existentially Optimal
[ Video , 35 min] - On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion
[ Video , 28 min] [ Slides ]
Publications
- How to Walk a Dog in Parallel: on Parallel Computation of the Discrete Fréchet Distance
Leonid Barenboim, Arnold Filtser, Omrit Filtser, Orr Fischer
To appear in DISC 2026.
The Fréchet distance (also known as the dog-leash distance) is a well-studied and prevalent distance measure for polygonal curves. In this paper we present parallel algorithms for computing the discrete Fréchet distance between two curves, in both the PRAM and MPC models.
For two polygonal curves of length $n$, we show that the distance between them can be computed in the PRAM model in $O(\log n/\sigma)$ rounds and $\tilde{O}(n^{2+\sigma})$ total work for any $0<\sigma<1$. In the sublinear MPC model with $n^\rho$ space per machine, it can be computed in $O(\log n/\rho^2)$ rounds and $\tilde{O}(n^{2+\rho})$ total space. For $c$-packed curves, we match our bounds to the sequential algorithms, showing that a $(1+\varepsilon)$-approximation of the distance can be computed in the PRAM model in $O(\log{n}+1/\sigma^3)$ rounds and $\tilde{O}(n^{1+\sigma})$ total work, and in the sublinear MPC model in $O(1/\rho^3)$ rounds and $\tilde{O}(n^{1+\rho})$ total space, for any constant $\varepsilon>0$. As part of our algorithms for $c$-packed curves, we also present parallel algorithms for computing curve simplifications, which may be of independent interest.
- Fatness and Flatness
Arnold Filtser, Hung Le, Nikolas Mählmann, Marcin Pilipczuk, Michał Pilipczuk
[ arXiv version ]Fat minors are the metric analog of graph minors that are tailored to the analysis of metric (edge-weighted) graphs and, more generally, metric spaces having a suitable notion of shortest paths. Despite a large interest in this notion, not much is known about the structure of metric graphs excluding a fixed fat minor.
We prove that if a metric graph $G$ excludes a fixed graph $H$ as a $δ$-fat minor, for some $δ>0$, then $G$ enjoys the metric analog of flatness (aka uniform quasi-wideness) - a structural property from the field of Sparsity. In essence, our flatness result says that for any $α\geq β$ large enough compared to $δ$, in every large enough set $A$ in $G$ one can find a sizable subset $B$ that becomes $α$-scattered after removing a bounded number of balls of radius $β$. We call this property drill-flatness. Notably, the proof only relies on excluding shallow fat minors: every branch set has radius at most $2α$.
As a corollary, we prove that metric graphs that exclude a fixed $δ$-fat minor have bounded $\varepsilon$-scatter dimension if we consider only $\varepsilon$-scatters at distances large enough compared to $δ$. By combining this with the results of Abbasi et al. [FOCS 2023], we infer that the $k$-Center problem on instances excluding $H$ as a $δ$-fat minor admits an approximation algorithm that finds a solution of cost at most $(1+\varepsilon)\cdot\mathsf{OPT}+{\cal O}(δ/\varepsilon^2)$ in time ${\cal O}_{H,\varepsilon}(n^{{\cal O}(1)})$. This is one of the first algorithmic results for general fat-minor-free metrics.
We also study drill-flatness in hereditary classes of (unweighted) graphs, where we obtain a characterization equating drill-flatness with excluding shallow induced minors. This is an induced analog of the equivalence between flatness and nowhere denseness - one of central results of Sparsity.
- Directed Low Diameter Decomposition for Structured Digraphs
Shinwoo An, Arnold Filtser
To appear in APPROX 2026.
[ arXiv version ]Low diameter decompositions, or LDDs for short, are a fundamental primitive in the design of efficient graph algorithms. Roughly speaking, an LDD is a distribution over partitions of the vertices into bounded-diameter clusters such that nearby vertices are likely to be clustered together. Recently, there has been growing interest in lifting the notion of LDDs into directed graphs. In particular, there are two natural directed analogues. The first is a directed LDD, where after removing a random subset of edges, every strongly connected component has a small diameter. The second is a quasipartition, which imposes the stronger requirement that whenever one vertex can still reach another after the edge removal, the two vertices must be close in the original directed metric. Every quasipartition yields an LDD, but the converse does not necessarily hold.
In this work, we initiate the systematic study of LDDs in structured directed graphs. As our first main result, we show that any directed graph with pathwidth $\mathsf{pw}$ admits an $(O(\mathsf{pw}), Δ)$-LDD. This improves upon the previous best-known $(2^{O(\mathsf{pw}^2)}, Δ)$-LDD construction, which was implicitly derived from the quasipartition result of Salmasi, Sidiropoulos, and Sridhar [SODA'19].
As our second result, we show that the integrality gap of the Directed Non-Bipartite Sparsest-Cut LP relaxation on an $n$-vertex graph with treewidth $\mathsf{tw}$ is $O(\mathsf{tw} \log n)$. This improves upon the $O(\mathsf{tw}\log^2 n)$ bound of Mémoli, Sidiropoulos, and Sridhar [ICALP'16, Algorithmica'18]. We obtain this result through the refined analysis of the quasipartition construction of Mémoli et al. for bounded treewidth graphs.
- Bi-Lipschitz extensions and outlier embeddings into trees
Shuchi Chawla, Arnold Filtser, Kristin Sheridan, Yoni Trachtenberg
To appear in APPROX 2026.
[ arXiv version ]We develop low distortion embeddings with outliers from arbitrary metrics into hierarchically separated trees (HSTs). In particular, we develop an efficient algorithm that for any $ε>0$, given an input metric $(X,d)$, and a probabilistic embedding of all but $k$ points from $X$ into HSTs with distortion $c$, samples from a probabilistic embedding of all but $O(\frac{k}ε\log k)$ points into HSTs that achieves distortion at most $(32+ε)c$.
Our results are based on two key technical components. First, we extend an algorithm of Munagala et al. [2023] for minimizing the distortion of embeddings without outliers into HSTs to the setting with outliers. We combine this with new results on bi-Lipschitz extensions into trees and $\ell_1$ space. In particular, we show that any probabilistic embedding into HSTs can be extended to $k$ additional points with only a factor $O(\log k)$ of additional distortion. This bi-Lipschitz extension result utilizes a new probabilistic partitioning scheme that we call onion partitioning.
- Near-Resolution of the Tradeoff Conjecture in Distributed Proof Labeling Schemes
Arnold Filtser, Orr Fischer
In PODC 2026.
[ arXiv version ] [ Conference version ]In the $t$-Proof Labeling Scheme model ($t$-PLS model), our goal is to certify that a network of nodes satisfies a given property $P$. A prover assigns a label to each node, and each node decides to accept or reject based on its labeled $t$-hop neighborhood. If $P$ holds, there exists a labeling that makes all nodes accept. If $P$ does not hold, in all labelings at least one node rejects. The cost of a scheme is its maximum label size.
The Tradeoff Conjecture [Feuilloley, Fraigniaud, Hirvonen, Paz, and Perry, DISC 18, Dist. Comput.~21] hypothesizes that the existence of a $1$-PLS for a property $P$ with cost $p$ implies the existence of a $t$-PLS for $P$ with cost $O(\lceil p/t \rceil)$. The conjecture was initially shown to hold for specific graph classes, such as trees, cycles, and grids. Later, a weaker $\widetilde{O}(\lceil Δp/\sqrt{t} \rceil)$ cost was shown for fixed minor-free graphs, where $Δ$ is the maximum degree.
In this work we resolve the Tradeoff Conjecture, up to a single logarithmic factor. In general graphs, we show that the existence of a $1$-PLS with cost $p$ implies the existence of an $O(t\log{n})$-PLS with cost $O(\lceil p/t \rceil)$ for the same property. For fixed minor-free graphs (which include e.g. planar graphs), we show that the existence of a $1$-PLS with cost $p$ implies the existence of a $t$-PLS with cost $O(\lceil p/t \rceil+\log{n})$ for the same property. We also refute a previously suggested stronger variant of the Tradeoff Conjecture, and show that having very large $t$-hop neighborhoods is an insufficient condition for obtaining a tradeoff better than $O(\lceil p/t \rceil)$.
- DAG Covers: The Steiner Point Effect
Sujoy Bhore, Hsien-Chih Chang, Jonathan Conroy, Arnold Filtser, Eunjin Oh, Nicole Wein, Da Wei Zheng
In ESA 2026.
[ arXiv version ] [ Conference version ]Given a weighted digraph $G$, a $(t,g,μ)$-DAG cover is a collection of $g$ dominating DAGs $D_1,\dots,D_g$ such that all distances are approximately preserved: for every pair $(u,v)$ of vertices, $\min_id_{D_i}(u,v)\le t\cdot d_{G}(u,v)$, and the total number of non-$G$ edges is bounded by $|(\cup_i D_i)\setminus G|\le μ$. Assadi, Hoppenworth, and Wein [STOC 25] and Filtser [SODA 26] studied DAG covers for general digraphs. This paper initiates the study of Steiner DAG cover, where the DAGs are allowed to contain Steiner points.
We obtain Steiner DAG covers on the important classes of planar digraphs and low-treewidth digraphs. Specifically, we show that any digraph with treewidth tw admits a $(1,2,\tilde{O}(n\cdot tw))$-Steiner DAG cover. For planar digraphs we provide a $(1+\varepsilon,2,\tilde{O}_\varepsilon(n))$-Steiner DAG cover.
We also demonstrate a stark difference between Steiner and non-Steiner DAG covers. As a lower bound, we show that any non-Steiner DAG cover for graphs with treewidth $1$ with stretch $t<2$ and sub-quadratic number of extra edges requires $Ω(\log n)$ DAGs.
- Dynamic Light Spanners for Doubling Metrics
Sujoy Bhore, Jonathan Conroy, Arnold Filtser
In SOCG 2026.
[ arXiv version ] [ Conference version ]A $t$-spanner of a point set $X$ in a metric space $(\mathcal{X}, δ)$ is a graph $G$ with vertex set $P$ such that, for any pair of points $u,v \in X$, the distance between $u$ and $v$ in $G$ is at most $t$ times $δ(u,v)$. We study the problem of maintaining a spanner for a dynamic point set $X$ -- that is, when $X$ undergoes a sequence of insertions and deletions -- in a metric space of constant doubling dimension. For any constant $\varepsilon>0$, we maintain a $(1+\varepsilon)$-spanner of $P$ whose total weight remains within a constant factor of the weight of the minimum spanning tree of $X$. Each update (insertion or deletion) can be performed in $\operatorname{poly}(\log Φ)$ time, where $Φ$ denotes the aspect ratio of $X$. Prior to our work, no efficient dynamic algorithm for maintaining a light-weight spanner was known even for point sets in low-dimensional Euclidean space.
- The Squishy Grid Problem
Zixi Cai, Kuowen Chen, Shengquan Du, Arnold Filtser, Seth Pettie, Daniel Skora
In SOCG 2026.
[ arXiv version ] [ Conference version ]In this paper we consider the problem of approximating Euclidean distances by the infinite integer grid graph. Although the topology of the graph is fixed, we have control over the edge-weight assignment $w:E\to \mathbb{R}_{\ge 0}$, and hope to have grid distances be asymptotically isometric to Euclidean distances, that is, for all grid points $u,v$, $\mathrm{dist}_w(u,v) = (1\pm o(1))\|u-v\|_2$. We give three methods for solving this problem, each attractive in its own way.
* Our first construction is based on an embedding of the recursive, non-periodic pinwheel tiling of Radin and Conway into the integer grid. Distances in the pinwheel graph are asymptotically isometric to Euclidean distances, but no explicit bound on the rate of convergence was known. We prove that the multiplicative distortion of the pinwheel graph is $(1+1/Θ(\log^ξ\log D))$, where $D$ is the Euclidean distance and $ξ=Θ(1)$. The pinwheel tiling approach is conceptually simple, but can be improved quantitatively.
* Our second construction is based on a hierarchical arrangement of "highways." It is simple, achieving stretch $(1 + 1/Θ(D^{1/9}))$, which converges doubly exponentially faster than the pinwheel tiling approach.
* The first two methods are deterministic. An even simpler approach is to sample the edge weights independently from a common distribution $\mathscr{D}$. Whether there exists a distribution $\mathscr{D}^*$ that makes grid distances Euclidean, asymptotically and in expectation, is major open problem in the theory of first passage percolation. Previous experiments show that when $\mathscr{D}$ is a Fisher distribution, grid distances are within 1% of Euclidean. We demonstrate experimentally that this level of accuracy can be achieved by a simple 2-point distribution that assigns weights 0.41 or 4.75 with probability 44% and 56%, respectively.
- FPT approximations for Capacitated Sum of Radii and Diameters
Arnold Filtser, Ameet Gadekar
In SOCG 2026.
[ arXiv version ] [ Conference version ]The Capacitated Sum of Radii problem involves partitioning a set of points $P$, where each point $p\in P$ has capacity $U_p$, into $k$ clusters that minimize the sum of cluster radii, such that the number of points in the cluster centered at point $p$ is at most $U_p$. We begin by showing that the problem is APX-hard, and that under gap-ETH there is no parameterized approximation scheme (FPT-AS). We then construct a $\approx5.83$-approximation algorithm in FPT time (improving a previous $\approx7.61$ approximation in FPT time). Our results also hold when the objective is a general monotone symmetric norm of radii. We also improve the approximation factors for the uniform capacity case, and for the closely related problem of Capacitated Sum of Diameters.
- Stochastic Embedding of Digraphs into DAGs
Arnold Filtser
In SODA 2026.
[ arXiv version ] [ Conference version ]Given a weighted digraph $G=(V,E,w)$, a stochastic embedding into DAGs is a distribution $\mathcal{D}$ over pairs of DAGs $(D_1,D_2)$ such that for every $u,v$: (1) the reachability is preserved: $u\rightsquigarrow_G v$ (i.e., $v$ is reachable from $u$ in $G$) implies that $u\rightsquigarrow_{D_1} v$ or $u\rightsquigarrow_{D_2} v$ (but not both), and (2) distances are dominated: $d_G(u,v)\le\min\{d_{D_1}(u,v),d_{D_2}(u,v)\}$. The stochastic embedding $\mathcal{D}$ has expected distortion $t$ if for every $u,v\in V$, \[ \mathbb{E}_{(D_{1},D_{2})\sim\mathcal{D}}\left[d_{D_{1}}(u,v)\cdot\boldsymbol{1}[u\rightsquigarrow_{D_{1}}v]+d_{D_{2}}(u,v)\cdot\boldsymbol{1}[u\rightsquigarrow_{D_{2}}v]\right]\le t\cdot d_{G}(u,v)~. \] Finally, the sparsity of $\mathcal{D}$ is the maximum number of edges in any of the DAGs in its support. Given an $n$ vertex digraph with $m$ edges, we construct a stochastic embedding into DAGs with expected distortion $\tilde{O}(\log n)$ and $\tilde{O}(m)$ sparsity, improving a previous result by Assadi, Hoppenworth, and Wein [STOC 25], which achieved expected distortion $\tilde{O}(\log^3 n)$. Further, we can sample DAGs from this distribution in $\tilde{O}(m)$ time.
- How to Protect Yourself from Threatening Skeletons: Optimal Padded Decompositions for Minor-Free Graphs
Jonathan Conroy, Arnold Filtser
In STOC 2025
[ arXiv version ] [ Conference version ] [ Video (STOC presentation by Jonathan Conroy)] [ Video (full length)]Roughly, a metric space has padding parameter $β$ if for every $Δ>0$, there is a stochastic decomposition of the metric points into clusters of diameter at most $Δ$ such that every ball of radius $γΔ$ is contained in a single cluster with probability at least $e^{-γβ}$. The padding parameter is an important characteristic of a metric space with vast algorithmic implications. In this paper we prove that the shortest path metric of every $K_r$-minor-free graph has padding parameter $O(\log r)$, which is also tight. This resolves a long standing open question, and exponentially improves the previous bound. En route to our main result, we construct sparse covers for $K_r$-minor-free graphs with improved parameters, and we prove a general reduction from sparse covers to padded decompositions.
- Faster Approximation Algorithms for k-Center via Data Reduction
Arnold Filtser, Shaofeng H.-C. Jiang, Yi Li, Anurag Murty Naredla, Ioannis Psarros, Qiaoyuan Yang, Qin Zhang
In ICML 2025
[ arXiv version ] [ Conference version ]We study efficient algorithms for the Euclidean $k$-Center problem, focusing on the regime of large $k$. We take the approach of data reduction by considering $α$-coreset, which is a small subset $S$ of the dataset $P$ such that any $β$-approximation on $S$ is an $(α+ β)$-approximation on $P$. We give efficient algorithms to construct coresets whose size is $k \cdot o(n)$, which immediately speeds up existing approximation algorithms. Notably, we obtain a near-linear time $O(1)$-approximation when $k = n^c$ for any $0 < c < 1$. We validate the performance of our coresets on real-world datasets with large $k$, and we observe that the coreset speeds up the well-known Gonzalez algorithm by up to $4$ times, while still achieving similar clustering cost. Technically, one of our coreset results is based on a new efficient construction of consistent hashing with competitive parameters. This general tool may be of independent interest for algorithm design in high dimensional Euclidean spaces.
- On Sparse Covers of Minor Free Graphs, Low Dimensional Metric Embeddings, and other applications
Arnold Filtser
In SOCG 2025
[ arXiv version ] [ Conference version ] [ Video (SOCG presentation)]Given a metric space $(X,d_X)$, a $(β,s,Δ)$-sparse cover is a collection of clusters $\mathcal{C}\subseteq P(X)$ with diameter at most $Δ$, such that for every point $x\in X$, the ball $B_X(x,\fracΔβ)$ is fully contained in some cluster $C\in \mathcal{C}$, and $x$ belongs to at most $s$ clusters in $\mathcal{C}$. Our main contribution is to show that the shortest path metric of every $K_r$-minor free graphs admits $(O(r),O(r^2),Δ)$-sparse cover, and for every $ε>0$, $(4+ε,O(\frac1ε)^r,Δ)$-sparse cover (for arbitrary $Δ>0$). We then use this sparse cover to show that every $K_r$-minor free graph embeds into $\ell_\infty^{\tilde{O}(\frac1ε)^{r+1}\cdot\log n}$ with distortion $3+ε$ (resp. into $\ell_\infty^{\tilde{O}(r^2)\cdot\log n}$ with distortion $O(r)$). Further, among other applications, this sparse cover immediately implies an algorithm for the oblivious buy-at-bulk problem in fixed minor free graphs with the tight approximation factor $O(\log n)$ (previously nothing beyond general graphs was known).
- Highway Dimension: a Metric View
Andreas Emil Feldmann, Arnold Filtser
In SODA 2025.
To appear in TheoretiCS 2026.
[ arXiv version ] [ Conference version ]Realistic metric spaces (such as road/transportation networks) tend to be much more algorithmically tractable than general metrics. In an attempt to formalize this intuition, Abraham et~al. (SODA 2010, JACM 2016) introduced the notion of highway dimension. A weighted graph $G$ has highway dimension $h$ if for every ball $B$ of radius $\approx4r$, there is a hitting set of size $h$ hitting all the shortest paths of length $>r$ in $B$. Unfortunately, this definition fails to incorporate some very natural metric spaces such as the grid graph, and the Euclidean plane.
We relax the definition of highway dimension by demanding to hit only approximate shortest paths. In addition to generalizing the original definition, this new definition also incorporates all doubling spaces (in particular the grid graph and the Euclidean plane). We then construct a PTAS for TSP under this new definition (improving a QPTAS w.r.t. the original more restrictive definition of Feldmann et~al. (SICOMP 2018)). Finally, we develop a basic metric toolkit for spaces with small highway dimension by constructing padded decompositions, sparse covers/partitions, and tree covers. An abundance of applications follow.
- Optimal Padded Decomposition For Bounded Treewidth Graphs
Arnold Filtser, Tobias Friedrich, Davis Issac, Nikhil Kumar, Hung Le, Nadym Mallek, Ziena Zeif
In TheoretiCS 2025.
[ arXiv version ] [ Journal version ]A $(β,δ,Δ)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $Δ$ such that for every vertex $v\in V$, the probability that $\rm{ball}_G(v,γΔ)$ is entirely contained in the cluster containing $v$ is at least $e^{-βγ}$ for every $γ\in [0,δ]$. Padded decompositions have been studied for decades and have found numerous applications, including metric embedding, multicommodity flow-cut gap, multicut, and zero extension problems, to name a few. In these applications, parameter $β$, called the padding parameter, is the most important parameter since it decides either the distortion or the approximation ratios. For general graphs with $n$ vertices, $β= Θ(\log n)$. Klein, Plotkin, and Rao showed that $K_r$-minor-free graphs have padding parameter $β= O(r^3)$, which is a significant improvement over general graphs when $r$ is a constant. A long-standing conjecture is to construct a padded decomposition for $K_r$-minor-free graphs with padding parameter $β= O(\log r)$. Despite decades of research, the best-known result is $β= O(r)$, even for graphs with treewidth at most $r$. In this work, we make significant progress toward the aforementioned conjecture by showing that graphs with treewidth $\rm{tw}$ admit a padded decomposition with padding parameter $O(\log \rm{tw})$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $O(\sqrt{ \log n \cdot \log(\rm{tw})})$ flow-cut gap, max flow-min multicut ratio of $O(\log(\rm{tw}))$, an $O(\log(\rm{tw}))$ approximation for the 0-extension problem, an $\ell^{O(\log n)}_\infty$ embedding with distortion $O(\log \rm{tw})$, and an $O(\log \rm{tw})$ bound for integrality gap for the uniform sparsest cut.
- Near-Optimal Approximate Fully-Dynamic All-Pairs Shortest Paths in Planar Graphs
Arnold Filtser, Gramoz Goranci, Neel Patel, Maximilian Probst Gutenberg
In FOCS 2024.
[ Conference version ]We study the fully-dynamic all-pair shortest paths (APSP) problem on planar graphs: given an $n$-vertex planar graph $G=(V,E)$ undergoing edge insertions and deletions, the goal is to efficiently process these updates and support distance and shortest path queries. We give a $(1+\epsilon)$-approximate dynamic algorithm that supports edge updates and distance queries in $n^{o(1)}$ time, for any $1/\operatorname{poly}(\log n)<\epsilon<1$. Our result is a significant improvement over the best previously known bound of $\tilde{O}(\sqrt{n})$ on update and query time due to [Abraham, Chechik, and Gavoille, STOC '12], and bypasses a $\Omega(\sqrt{n})$ conditional lower-bound on update and query time for exact fully dynamic planar APSP [Abboud and Dahlgaard, FOCS '16]. The main technical contribution behind our result is to dynamize the planar emulator construction due to [Chang, Krauthgamer, Tan, STOC '22].
- Light, Reliable Spanners
Arnold Filtser, Yuval Gitlitz, Ofer Neiman
In SOCG 2024
[ arXiv version ] [ Conference version ]A $ν$-reliable spanner of a metric space $(X,d)$, is a (dominating) graph $H$, such that for any possible failure set $B\subseteq X$, there is a set $B^+$ just slightly larger $|B^+|\le(1+ν)\cdot|B|$, and all distances between pairs in $X\setminus B^+$ are (approximately) preserved in $H\setminus B$. Recently, there have been several works on sparse reliable spanners in various settings, but so far, the weight of such spanners has not been analyzed at all. In this work, we initiate the study of light reliable spanners, whose weight is proportional to that of the Minimum Spanning Tree (MST) of $X$.
We first observe that unlike sparsity, the lightness of any deterministic reliable spanner is huge, even for the metric of the simple path graph. Therefore, randomness must be used: an oblivious reliable spanner is a distribution over spanners, and the bound on $|B^+|$ holds in expectation.
We devise an oblivious $ν$-reliable $(2+\frac{2}{k-1})$-spanner for any $k$-HST, whose lightness is $\approx ν^{-2}$. We demonstrate a matching $Ω(ν^{-2})$ lower bound on the lightness (for any finite stretch). We also note that any stretch below 2 must incur linear lightness.
For general metrics, doubling metrics, and metrics arising from minor-free graphs, we construct light tree covers, in which every tree is a $k$-HST of low weight. Combining these covers with our results for $k$-HSTs, we obtain oblivious reliable light spanners for these metric spaces, with nearly optimal parameters. In particular, for doubling metrics we get an oblivious $ν$-reliable $(1+\varepsilon)$-spanner with lightness $\varepsilon^{-O({\rm ddim})}\cdot\tilde{O}(ν^{-2}\cdot\log n)$, which is best possible (up to lower order terms).
- Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings
Sujoy Bhore, Arnold Filtser, Csaba D. Tóth
In SODA 2024.
[ arXiv version ] [ Conference version ]Low-distortional metric embeddings are a crucial component in the modern algorithmic toolkit. In an online metric embedding, points arrive sequentially and the goal is to embed them into a simple space irrevocably, while minimizing the distortion. Our first result is a deterministic online embedding of a general metric into Euclidean space with distortion $O(\log n)\cdot\min\{\sqrt{\logΦ},\sqrt{n}\}$ (or, $O(d)\cdot\min\{\sqrt{\logΦ},\sqrt{n}\}$ if the metric has doubling dimension $d$), solving a conjecture by Newman and Rabinovich (2020), and quadratically improving the dependence on the aspect ratio $Φ$ from Indyk et al. (2010). Our second result is a stochastic embedding of a metric space into trees with expected distortion $O(d\cdot \logΦ)$, generalizing previous results (Indyk et al. (2010), Bartal et al. (2020)).
Next, we study the online minimum-weight perfect matching problem, where a sequence of $2n$ metric points arrive in pairs, and one has to maintain a perfect matching at all times. We allow recourse (as otherwise the order of arrival determines the matching). The goal is to return a perfect matching that approximates the minimum-weight perfect matching at all times, while minimizing the recourse. Our third result is a randomized algorithm with competitive ratio $O(d\cdot \log Φ)$ and recourse $O(\log Φ)$ against an oblivious adversary, this result is obtained via our new stochastic online embedding. Our fourth result is a deterministic algorithm against an adaptive adversary, using $O(\log^2 n)$ recourse, that maintains a matching of weight at most $O(\log n)$ times the weight of the MST, i.e., a matching of lightness $O(\log n)$. We complement our upper bounds with a strategy for an oblivious adversary that, with recourse $r$, establishes a lower bound of $Ω(\frac{\log n}{r \log r})$ for both competitive ratio and lightness.
- One Tree to Rule Them All: Poly-Logarithmic Universal Steiner Tree
Costas Busch, Da Qi Chen, Arnold Filtser, Daniel Hathcock, D Ellis Hershkowitz, Rajmohan Rajaraman
In FOCS 2023.
[ arXiv version ] [ Conference version ] [ Video (Talk by Ellis Hershkowitz)] [ Video (Talk on sparse partitions)]A spanning tree $T$ of graph $G$ is a $ρ$-approximate universal Steiner tree (UST) for root vertex $r$ if, for any subset of vertices $S$ containing $r$, the cost of the minimal subgraph of $T$ connecting $S$ is within a $ρ$ factor of the minimum cost tree connecting $S$ in $G$. Busch et al. (FOCS 2012) showed that every graph admits $2^{O(\sqrt{\log n})}$-approximate USTs by showing that USTs are equivalent to strong sparse partition hierarchies (up to poly-logs). Further, they posed poly-logarithmic USTs and strong sparse partition hierarchies as open questions.
We settle these open questions by giving polynomial-time algorithms for computing both $O(\log ^ 7 n)$-approximate USTs and poly-logarithmic strong sparse partition hierarchies. For graphs with constant doubling dimension or constant pathwidth we improve this to $O(\log n)$-approximate USTs and $O(1)$ strong sparse partition hierarchies. Our doubling dimension result is tight up to second order terms. We reduce the existence of these objects to the previously studied cluster aggregation problem and what we call dangling nets.
- Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings
Arnold Filtser
In SOCG 2023.
[ arXiv version ] [ Conference version ] [ Video (SOCG presentation)]Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive orderings (LSO) for Euclidean space. A $(τ,ρ)$-LSO is a collection $Σ$ of orderings such that for every $x,y\in\mathbb{R}^d$ there is an ordering $σ\inΣ$, where all the points between $x$ and $y$ w.r.t. $σ$ are in the $ρ$-neighborhood of either $x$ or $y$. In essence, LSO allow one to reduce problems to the $1$-dimensional line. Later, Filtser and Le [STOC 2022] developed LSO's for doubling metrics, general metric spaces, and minor free graphs. For Euclidean and doubling spaces, the number of orderings in the LSO is exponential in the dimension, which made them mainly useful for the low dimensional regime. In this paper, we develop new LSO's for Euclidean, $\ell_p$, and doubling spaces that allow us to trade larger stretch for a much smaller number of orderings. We then use our new LSO's (as well as the previous ones) to construct path reporting low hop spanners, fault tolerant spanners, reliable spanners, and light spanners for different metric spaces. While many nearest neighbor search (NNS) data structures were constructed for metric spaces with implicit distance representations (where the distance between two metric points can be computed using their names, e.g. Euclidean space), for other spaces almost nothing is known. In this paper we initiate the study of the labeled NNS problem, where one is allowed to artificially assign labels (short names) to metric points. We use LSO's to construct efficient labeled NNS data structures in this model.
- Streaming Facility Location in High Dimension via Geometric Hashing
Artur Czumaj, Arnold Filtser, Shaofeng H.-C. Jiang, Robert Krauthgamer, Pavel Veselý, Mingwei Yang
[ arXiv version ]In Euclidean Uniform Facility Location (UFL), the input is a set of clients in $\mathbb{R}^d$ and the goal is to place facilities to serve them, so as to minimize the total cost of opening facilities plus connecting the clients. We study the setting of dynamic geometric streams, where the clients are presented as a sequence of insertions and deletions of points in the grid $\{1,\ldots,Δ\}^d$, and we focus on the high-dimensional regime, where the algorithm must use space polynomial in $d\cdot\logΔ$.
We present a new algorithmic framework, based on importance sampling, for $O(1)$-approximation of UFL using only $\mathrm{poly}(d\cdot\logΔ)$ space. This framework is easy to implement in two passes, one for sampling points and the other for estimating their contribution. Over random-order streams, we can extend this to one pass by using the two halves of the stream separately. Our main result, for arbitrary-order streams, computes $O(d / \log d)$-approximation in one pass by combining the two passes differently. This improves upon previous algorithms that either need space $\exp(d)$ or only guarantee $O(d\cdot\log^2Δ)$-approximation, and therefore our algorithms for high dimension are the first to avoid the $O(\logΔ)$-factor in approximation that is inherent to the widely-used quadtree decomposition. Our improvement is achieved by employing a geometric hashing scheme that maps points in $\mathbb{R}^d$ into buckets of bounded diameter, with the key property that every point set of small-enough diameter is hashed into few buckets. By applying an alternative bound for this hashing, we also obtain an $O(1 / ε)$-approximation in one pass, using larger but still sublinear space $O(n^ε)$ where $n$ is the number of clients.
We complement our results by showing $1.085$-approximation requires space exponential in $\mathrm{poly}(d\cdot\logΔ)$.
- Expander Decomposition in Dynamic Streams
Arnold Filtser, Michael Kapralov, Mikhail Makarov
In ITCS 2023.
[ arXiv version ] [ Conference version ] [ Video (ITCS presentation by Mikhail Makarov)]In this paper we initiate the study of expander decompositions of a graph $G=(V, E)$ in the streaming model of computation. The goal is to find a partitioning $\mathcal{C}$ of vertices $V$ such that the subgraphs of $G$ induced by the clusters $C \in \mathcal{C}$ are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model.
Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of $V$) to within a $(δ, ε)$-multiplicative/additive error with high probability. The power cut sparsifier uses $\tilde{O}(n/εδ)$ space and edges, which we show is asymptotically tight up to polylogarithmic factors in $n$ for constant $δ$.
- Communication Complexity of Inner Product in Symmetric Normed Spaces
Alexandr Andoni, Jarosław Błasiok, Arnold Filtser
In ITCS 2023.
[ arXiv version ] [ Conference version ]We introduce and study the communication complexity of computing the inner product of two vectors, where the input is restricted w.r.t. a norm $N$ on the space $\mathbb{R}^n$. Here, Alice and Bob hold two vectors $v,u$ such that $\|v\|_N\le 1$ and $\|u\|_{N^*}\le 1$, where $N^*$ is the dual norm. They want to compute their inner product $\langle v,u \rangle$ up to an $\varepsilon$ additive term. The problem is denoted by $\mathrm{IP}_N$.
We systematically study $\mathrm{IP}_N$, showing the following results:
- For any symmetric norm $N$, given $\|v\|_N\le 1$ and $\|u\|_{N^*}\le 1$ there is a randomized protocol for $\mathrm{IP}_N$ using $\tilde{\mathcal{O}}(\varepsilon^{-6} \log n)$ bits -- we will denote this by $\mathcal{R}_{\varepsilon,1/3}(\mathrm{IP}_{N}) \leq \tilde{\mathcal{O}}(\varepsilon^{-6} \log n)$.
- One way communication complexity $\overrightarrow{\mathcal{R}}(\mathrm{IP}_{\ell_p})\leq\mathcal{O}(\varepsilon^{-\max(2,p)}\cdot \log\frac n\varepsilon)$, and a nearly matching lower bound $\overrightarrow{\mathcal{R}}(\mathrm{IP}_{\ell_p}) \geq Ω(\varepsilon^{-\max(2,p)})$ for $\varepsilon^{-\max(2,p)} \ll n$.
- One way communication complexity $\overrightarrow{\mathcal{R}}(N)$ for a symmetric norm $N$ is governed by embeddings $\ell_\infty^k$ into $N$. Specifically, while a small distortion embedding easily implies a lower bound $Ω(k)$, we show that, conversely, non-existence of such an embedding implies protocol with communication $k^{\mathcal{O}(\log \log k)} \log^2 n$.
- For arbitrary origin symmetric convex polytope $P$, we show $\mathcal{R}(\mathrm{IP}_{N}) \le\mathcal{O}(\varepsilon^{-2} \log \mathrm{xc}(P))$, where $N$ is the unique norm for which $P$ is a unit ball, and $\mathrm{xc}(P)$ is the extension complexity of $P$.
- Low Treewidth Embeddings of Planar and Minor-Free Metrics
Arnold Filtser, Hung Le
In FOCS 2022.
[ arXiv version ] [ Conference version ] [ Video (FOCS presentation by Hung Le)]Cohen-Addad, Filtser, Klein and Le [FOCS'20] constructed a stochastic embedding of minor-free graphs of diameter $D$ into graphs of treewidth $O_ε(\log n)$ with expected additive distortion $+εD$. Cohen-Addad et al. then used the embedding to design the first quasi-polynomial time approximation scheme (QPTAS) for the capacitated vehicle routing problem. Filtser and Le [STOC'21] used the embedding (in a different way) to design a QPTAS for the metric Baker's problems in minor-free graphs. In this work, we devise a new embedding technique to improve the treewidth bound of Cohen-Addad et al. exponentially to $O_ε(\log\log n)^2$. As a corollary, we obtain the first efficient PTAS for the capacitated vehicle routing problem in minor-free graphs. We also significantly improve the running time of the QPTAS for the metric Baker's problems in minor-free graphs from $n^{O_ε(\log(n))}$ to $n^{O_ε(\log\log(n))^3}$.
Applying our embedding technique to planar graphs, we obtain a deterministic embedding of planar graphs of diameter $D$ into graphs of treewidth $O((\log\log n)^2)/ε)$ and additive distortion $+εD$ that can be constructed in nearly linear time. Important corollaries of our result include a bicriteria PTAS for metric Baker's problems and a PTAS for the vehicle routing problem with bounded capacity in planar graphs, both run in almost-linear time. The running time of our algorithms is significantly better than previous algorithms that require quadratic time.
A key idea in our embedding is the construction of an (exact) emulator for tree metrics with treewidth $O(\log\log n)$ and hop-diameter $O(\log \log n)$. This result may be of independent interest.
- Online Spanners in Metric Spaces
Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, Csaba D. Tóth
In ESA 2022.
In SIDMA 2024.
[ arXiv version ] [ Journal version ] [ Conference version ]Given a metric space $\mathcal{M}=(X,δ)$, a weighted graph $G$ over $X$ is a metric $t$-spanner of $\mathcal{M}$ if for every $u,v \in X$, $δ(u,v)\le d_G(u,v)\le t\cdot δ(u,v)$, where $d_G$ is the shortest path metric in $G$. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points $(s_1, \ldots, s_n)$, where the points are presented one at a time (i.e., after $i$ steps, we saw $S_i = \{s_1, \ldots , s_i\}$). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a $t$-spanner $G_i$ for $S_i$ for all $i$, while minimizing the number of edges, and their total weight.
We construct online $(1+\varepsilon)$-spanners in Euclidean $d$-space, $(2k-1)(1+\varepsilon)$-spanners for general metrics, and $(2+\varepsilon)$-spanners for ultrametrics. Most notably, in Euclidean plane, we construct a $(1+\varepsilon)$-spanner with competitive ratio $O(\varepsilon^{-3/2}\log\varepsilon^{-1}\log n)$, bypassing the classic lower bound $Ω(\varepsilon^{-2})$ for lightness, which compares the weight of the spanner, to that of the MST.
- Locality-Sensitive Orderings and Applications to Reliable Spanners
Arnold Filtser, Hung Le
In STOC 2022.
[ arXiv version ] [ Conference version ] [ Video (STOC presentation)] [ Video (IDEAL workshop)]Chan, Har-Peled, and Jones [2020] recently developed locality-sensitive ordering (LSO), a new tool that allows one to reduce problems in the Euclidean space $\mathbb{R}^d$ to the $1$-dimensional line. They used LSO's to solve a host of problems. Later, Buchin, Har-Peled, and Oláh [2019,2020] used the LSO of Chan et al. to construct very sparse reliable spanners for the Euclidean space. A highly desirable feature of a reliable spanner is its ability to withstand a massive failure: the network remains functioning even if 90% of the nodes fail. In a follow-up work, Har-Peled, Mendel, and Oláh [2021] constructed reliable spanners for general and topologically structured metrics. Their construction used a different approach, and is based on sparse covers.
In this paper, we develop the theory of LSO's in non-Euclidean metrics by introducing new types of LSO's suitable for general and topologically structured metrics. We then construct such LSO's, as well as constructing considerably improved LSO's for doubling metrics. Afterwards, we use our new LSO's to construct reliable spanners with improved stretch and sparsity parameters. Most prominently, we construct $\tilde{O}(n)$-size reliable spanners for trees and planar graphs with the optimal stretch of $2$. Along the way to the construction of LSO's and reliable spanners, we introduce and construct ultrametric covers, and construct $2$-hop reliable spanners for the line.
- Hop-Constrained Metric Embeddings and their Applications
Arnold Filtser
In FOCS 2021.
[ arXiv version ] [ Conference version ] [ Video (FOCS presentation)]In network design problems, such as compact routing, the goal is to route packets between nodes using the (approximated) shortest paths. A desirable property of these routes is a small number of hops, which makes them more reliable, and reduces the transmission costs. Following the overwhelming success of stochastic tree embeddings for algorithmic design, Haeupler, Hershkowitz, and Zuzic (STOC'21) studied hop-constrained Ramsey-type metric embeddings into trees. Specifically, embedding $f:G(V,E)\rightarrow T$ has Ramsey hop-distortion $(t,M,β,h)$ (here $t,β,h\ge1$ and $M\subseteq V$) if $\forall u,v\in M$, $d_G^{(β\cdot h)}(u,v)\le d_T(u,v)\le t\cdot d_G^{(h)}(u,v)$. $t$ is called the distortion, $β$ is called the hop-stretch, and $d_G^{(h)}(u,v)$ denotes the minimum weight of a $u-v$ path with at most $h$ hops. Haeupler et al. constructed embedding where $M$ contains $1-ε$ fraction of the vertices and $β=t=O(\frac{\log^2 n}ε)$. They used their embedding to obtain multiple bicriteria approximation algorithms for hop-constrained network design problems.
In this paper, we first improve the Ramsey-type embedding to obtain parameters $t=β=\frac{\tilde{O}(\log n)}ε$, and generalize it to arbitrary distortion parameter $t$ (in the cost of reducing the size of $M$). This embedding immediately implies polynomial improvements for all the approximation algorithms from Haeupler et al.. Further, we construct hop-constrained clan embeddings (where each vertex has multiple copies), and use them to construct bicriteria approximation algorithms for the group Steiner tree problem, matching the state of the art of the non constrained version. Finally, we use our embedding results to construct hop constrained distance oracles, distance labeling, and most prominently, the first hop constrained compact routing scheme with provable guarantees.
- Clan Embeddings into Trees, and Low Treewidth Graphs
Arnold Filtser, Hung Le
In STOC 2021.
[ arXiv version ] [ Conference version ] [ Video (STOC presentation)] [ Video (in Hebrew)]In low distortion metric embeddings, the goal is to embed a host "hard" metric space into a "simpler" target space while approximately preserving pairwise distances. A highly desirable target space is that of a tree metric. Unfortunately, such embedding will result in a huge distortion. A celebrated bypass to this problem is stochastic embedding with logarithmic expected distortion. Another bypass is Ramsey-type embedding, where the distortion guarantee applies only to a subset of the points. However, both these solutions fail to provide an embedding into a single tree with a worst-case distortion guarantee on all pairs. In this paper, we propose a novel third bypass called clan embedding. Here each point $x$ is mapped to a subset of points $f(x)$, called a clan, with a special chief point $χ(x)\in f(x)$. The clan embedding has multiplicative distortion $t$ if for every pair $(x,y)$ some copy $y'\in f(y)$ in the clan of $y$ is close to the chief of $x$: $\min_{y'\in f(y)}d(y',χ(x))\le t\cdot d(x,y)$. Our first result is a clan embedding into a tree with multiplicative distortion $O(\frac{\log n}ε)$ such that each point has $1+ε$ copies (in expectation). In addition, we provide a "spanning" version of this theorem for graphs and use it to devise the first compact routing scheme with constant size routing tables.
We then focus on minor-free graphs of diameter prameterized by $D$, which were known to be stochastically embeddable into bounded treewidth graphs with expected additive distortion $εD$. We devise Ramsey-type embedding and clan embedding analogs of the stochastic embedding. We use these embeddings to construct the first (bicriteria quasi-polynomial time) approximation scheme for the metric $ρ$-dominating set and metric $ρ$-independent set problems in minor-free graphs.
- Condorcet Relaxation in Spatial Voting (or Plurality in Spatial Voting Games with Constant β)
Arnold Filtser, Omrit Filtser
In AAAI 2021.
In DCG 2023.
[ arXiv version ] [ Journal version ] [ Conference version ]Consider a set $V$ of voters, represented by a multiset in a metric space $(X,d)$. The voters have to reach a decision -- a point in $X$. A choice $p\in X$ is called a $β$-plurality point for $V$, if for any other choice $q\in X$ it holds that $|\{v\in V\mid β\cdot d(p,v)\le d(q,v)\}|\ge\frac{|V|}{2}$. In other words, at least half of the voters ``prefer'' $p$ over $q$, when an extra factor of $β$ is taken in favor of $p$. For $β=1$, this is equivalent to Condorcet winner, which rarely exists. The concept of $β$-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [TALG 2021] as a relaxation of the Condorcet criterion.
Let $β^*_{(X,d)}=\sup\{β\mid \mbox{every finite multiset $V$ in $X$ admits a $β$-plurality point}\}$. The parameter $β^*$ determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane $β^*_{(\mathbb{R}^2,\|\cdot\|_2)}=\frac{\sqrt{3}}{2}$, and more generally, for $d$-dimensional Euclidean space, $\frac{1}{\sqrt{d}}\le β^*_{(\mathbb{R}^d,\|\cdot\|_2)}\le\frac{\sqrt{3}}{2}$. In this paper, we show that $0.557\le β^*_{(\mathbb{R}^d,\|\cdot\|_2)}$ for any dimension $d$ (notice that $\frac{1}{\sqrt{d}}<0.557$ for any $d\ge 4$). In addition, we prove that for every metric space $(X,d)$ it holds that $\sqrt{2}-1\leβ^*_{(X,d)}$, and show that there exists a metric space for which $β^*_{(X,d)}\le \frac12$.
- Graph Spanners by Sketching in Dynamic Streams and the Simultaneous Communication Model
Arnold Filtser, Michael Kapralov, Navid Nouri
In SODA 2021.
[ arXiv version ] [ Conference version ] [ Video (SODA presentation)]Graph sketching is a powerful technique introduced by the seminal work of Ahn, Guha and McGregor'12 on connectivity in dynamic graph streams that has enjoyed considerable attention in the literature since then, and has led to near optimal dynamic streaming algorithms for many fundamental problems such as connectivity, cut and spectral sparsifiers and matchings. Interestingly, however, the sketching and dynamic streaming complexity of approximating the shortest path metric of a graph is still far from well-understood. Besides a direct $k$-pass implementation of classical spanner constructions (recently improved to $\lfloor\frac k2\rfloor+1$-passes by Fernandez, Woodruff and Yasuda'20) the state of the art amounts to a $O(\log k)$-pass algorithm of Ahn, Guha and McGregor'12, and a $2$-pass algorithm of Kapralov and Woodruff'14. In particular, no single pass algorithm is known, and the optimal tradeoff between the number of passes, stretch and space complexity is open.
In this paper we introduce several new graph sketching techniques for approximating the shortest path metric of the input graph. We give the first single pass sketching algorithm for constructing graph spanners: we show how to obtain a $\widetilde{O}(n^{\frac23})$-spanner using $\widetilde{O}(n)$ space, and in general a $\widetilde{O}(n^{\frac23(1-α)})$-spanner using $\widetilde{O}(n^{1+α})$ space for every $α\in [0, 1]$, a tradeoff that we think may be close optimal. We also give new spanner construction algorithms for any number of passes, simultaneously improving upon all prior work on this problem. Finally, we study the simultaneous communication model and propose the first protocols with low per player information.
- Static and Streaming Data Structures for Fréchet Distance Queries
Arnold Filtser, Omrit Filtser
In SODA 2021.
In TALG 2023.
[ arXiv version ] [ Journal version ] [ Conference version ] [ Video (SODA presentation by Omrit Filtser)]Given a curve $P$ with points in $\mathbb{R}^d$ in a streaming fashion, and parameters $\varepsilon>0$ and $k$, we construct a distance oracle that uses $O(\frac{1}{\varepsilon})^{kd}\log\varepsilon^{-1}$ space, and given a query curve $Q$ with $k$ points in $\mathbb{R}^d$, returns in $\tilde{O}(kd)$ time a $1+\varepsilon$ approximation of the discrete Fréchet distance between $Q$ and $P$.
In addition, we construct simplifications in the streaming model, oracle for distance queries to a sub-curve (in the static setting), and introduce the zoom-in problem. Our algorithms work in any dimension $d$, and therefore we generalize some useful tools and algorithms for curves under the discrete Fréchet distance to work efficiently in high dimensions.
- On light spanners, low-treewidth embeddings and efficient traversing in minor-free graphs
Vincent Cohen-Addad, Arnold Filtser, Philip N. Klein, Hung Le
In FOCS 2020.
[ arXiv version ] [ Conference version ] [ Video (FOCS presentation by Hung Le)]Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A fundamental idea that helps both to understand the structural properties of these metrics and lead to strong algorithmic results is to construct a "small-complexity" graph that approximately preserves distances between pairs of points of the metric. We show the two following structural results for minor-free metrics:
1. Construction of a light subset spanner. Given a subset of vertices called terminals, and $ε$, in polynomial time we construct a subgraph that preserves all pairwise distances between terminals up to a multiplicative $1+ε$ factor, of total weight at most $O_ε(1)$ times the weight of the minimal Steiner tree spanning the terminals.
2. Construction of a stochastic metric embedding into low treewidth graphs with expected additive distortion $εD$. Namely, given a minor free graph $G=(V,E,w)$ of diameter $D$, and parameter $ε$, we construct a distribution $\mathcal{D}$ over dominating metric embeddings into treewidth-$O_ε(\log n)$ graphs such that the additive distortion is at most $εD$.
One of our important technical contributions is a novel framework that allows us to reduce both problems to problems on simpler graphs of bounded diameter. Our results have the following algorithmic consequences: (1) the first efficient approximation scheme for subset TSP in minor-free metrics; (2) the first approximation scheme for vehicle routing with bounded capacity in minor-free metrics; (3) the first efficient approximation scheme for vehicle routing with bounded capacity on bounded genus metrics.
- Distributed Construction of Light Networks
Michael Elkin, Arnold Filtser, Ofer Neiman
In PODC 2020.
[ arXiv version ] [ Conference version ] [ Video (PODC presentation)]A $t$-spanner $H$ of a weighted graph $G=(V,E,w)$ is a subgraph that approximates all pairwise distances up to a factor of $t$. The lightness of $H$ is defined as the ratio between the weight of $H$ to that of the minimum spanning tree. An $(α,β)$-Shallow Light Tree (SLT) is a tree of lightness $β$, that approximates all distances from a designated root vertex up to a factor of $α$. A long line of works resulted in efficient algorithms that produce (nearly) optimal light spanners and SLTs.
Some of the most notable algorithmic applications of light spanners and SLTs are in distributed settings. Surprisingly, so far there are no known efficient distributed algorithms for constructing these objects in general graphs. In this paper we devise efficient distributed algorithms in the CONGEST model for constructing light spanners and SLTs, with near optimal parameters. Specifically, for any $k\ge 1$ and $0<ε<1$, we show a $(2k-1)\cdot(1+ε)$-spanner with lightness $O(k\cdot n^{1/k})$ can be built in $\tilde{O}\left(n^{\frac12+\frac{1}{4k+2}}+D\right)$ rounds (where $n=|V|$ and $D$ is the hop-diameter of $G$). In addition, for any $α>1$ we provide an $(α,1+\frac{O(1)}{α-1})$-SLT in $(\sqrt{n}+D)\cdot n^{o(1)}$ rounds. The running time of our algorithms cannot be substantially improved.
We also consider spanners for the family of doubling graphs, and devise a $(\sqrt{n}+D)\cdot n^{o(1)}$ rounds algorithm in the CONGEST model that computes a $(1+ε)$-spanner with lightness $(\log n)/ε^{O(1)}$. As a stepping stone, which is interesting in its own right, we first develop a distributed algorithm for constructing nets (for arbitrary weighted graphs), generalizing previous algorithms that worked only for unweighted graphs.
- Scattering and Sparse Partitions, and their Applications
Arnold Filtser
In ICALP 2020.
In TALG 2024.
[ arXiv version ] [ Journal version ] [ Conference version ] [ Video 1] [ Video 2] [ Video 3 (ICALP presentation)]A partition $\mathcal{P}$ of a weighted graph $G$ is $(σ,τ,Δ)$-sparse if every cluster has diameter at most $Δ$, and every ball of radius $Δ/σ$ intersects at most $τ$ clusters. Similarly, $\mathcal{P}$ is $(σ,τ,Δ)$-scattering if instead for balls we require that every shortest path of length at most $Δ/σ$ intersects at most $τ$ clusters. Given a graph $G$ that admits a $(σ,τ,Δ)$-sparse partition for all $Δ>0$, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch $O(τσ^2\log_τn)$. Given a graph $G$ that admits a $(σ,τ,Δ)$-scattering partition for all $Δ>0$, we construct a solution for the Steiner Point Removal problem with stretch $O(τ^3σ^3)$. We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.
- Approximate Nearest Neighbor for Curves --- Simple, Efficient, and Deterministic
Arnold Filtser, Omrit Filtser, Matthew J. Katz
In ICALP 2020.
In Algorithmica 2023.
[ arXiv version ] [ Journal version ] [ Conference version ] [ Video (ICALP presentation by Omrit Filtser)]In the $(1+\varepsilon,r)$-approximate near-neighbor problem for curves (ANNC) under some distance measure $δ$, the goal is to construct a data structure for a given set $\mathcal{C}$ of curves that supports approximate near-neighbor queries: Given a query curve $Q$, if there exists a curve $C\in\mathcal{C}$ such that $δ(Q,C)\le r$, then return a curve $C'\in\mathcal{C}$ with $δ(Q,C')\le(1+\varepsilon)r$. There exists an efficient reduction from the $(1+\varepsilon)$-approximate nearest-neighbor problem to ANNC, where in the former problem the answer to a query is a curve $C\in\mathcal{C}$ with $δ(Q,C)\le(1+\varepsilon)\cdotδ(Q,C^*)$, where $C^*$ is the curve of $\mathcal{C}$ closest to $Q$. Given a set $\mathcal{C}$ of $n$ curves, each consisting of $m$ points in $d$ dimensions, we construct a data structure for ANNC that uses $n\cdot O(\frac{1}{\varepsilon})^{md}$ storage space and has $O(md)$ query time (for a query curve of length $m$), where the similarity between two curves is their discrete Fréchet or dynamic time warping distance. Our method is simple to implement, deterministic, and results in an exponential improvement in both query time and storage space compared to all previous bounds. Further, we also consider the asymmetric version of ANNC, where the length of the query curves is $k \ll m$, and obtain essentially the same storage and query bounds as above, except that $m$ is replaced by $k$. Finally, we apply our method to a version of approximate range counting for curves and achieve similar bounds.
- Labelings vs. Embeddings: On Distributed Representations of Distances
Arnold Filtser, Lee-Ad Gottlieb, Robert Krauthgamer
In SODA 2020.
In DCG 2024.
[ arXiv version ] [ Journal version ] [ Conference version ]We investigate for which metric spaces the performance of distance labeling and of $\ell_\infty$-embeddings differ, and how significant can this difference be. Recall that a distance labeling is a distributed representation of distances in a metric space $(X,d)$, where each point $x\in X$ is assigned a succinct label, such that the distance between any two points $x,y \in X$ can be approximated given only their labels. A highly structured special case is an embedding into $\ell_\infty$, where each point $x\in X$ is assigned a vector $f(x)$ such that $\|f(x)-f(y)\|_\infty$ is approximately $d(x,y)$. The performance of a distance labeling or an $\ell_\infty$-embedding is measured via its distortion and its label-size/dimension.
We also study the analogous question for the prioritized versions of these two measures. Here, a priority order $π=(x_1,\dots,x_n)$ of the point set $X$ is given, and higher-priority points should have shorter labels. Formally, a distance labeling has prioritized label-size $α(\cdot)$ if every $x_j$ has label size at most $α(j)$. Similarly, an embedding $f: X \to \ell_\infty$ has prioritized dimension $α(\cdot)$ if $f(x_j)$ is non-zero only in the first $α(j)$ coordinates. In addition, we compare these prioritized measures to their classical (worst-case) versions.
We answer these questions in several scenarios, uncovering a surprisingly diverse range of behaviors. First, in some cases labelings and embeddings have very similar worst-case performance, but in other cases there is a huge disparity. However in the prioritized setting, we most often find a strict separation between the performance of labelings and embeddings. And finally, when comparing the classical and prioritized settings, we find that the worst-case bound for label size often "translates" to a prioritized one, but also find a surprising exception to this rule.
- A Face Cover Perspective to ℓ1 Embeddings of Planar Graphs
Arnold Filtser
In SODA 2020.
In TALG 2025.
[ arXiv version ] [ Journal version ] [ Conference version ] [ Video]It was conjectured by Gupta et al. [Combinatorica04] that every planar graph can be embedded into $\ell_1$ with constant distortion. However, given an $n$-vertex weighted planar graph, the best upper bound on the distortion is only $O(\sqrt{\log n})$, by Rao [SoCG99]. In this paper we study the case where there is a set $K$ of terminals, and the goal is to embed only the terminals into $\ell_1$ with low distortion. In a seminal paper, Okamura and Seymour [J.Comb.Theory81] showed that if all the terminals lie on a single face, they can be embedded isometrically into $\ell_1$. The more general case, where the set of terminals can be covered by $γ$ faces, was studied by Lee and Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the art is an upper bound of $O(\log γ)$ by Krauthgamer, Lee and Rika [SODA19]. Our contribution is a further improvement on the upper bound to $O(\sqrt{\logγ})$. Since every planar graph has at most $O(n)$ faces, any further improvement on this result, will be a major breakthrough, directly improving upon Rao's long standing upper bound. Moreover, it is well known that the flow-cut gap equals to the distortion of the best embedding into $\ell_1$. Therefore, our result provides a polynomial time $O(\sqrt{\log γ})$-approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by $γ$ faces.
- On Strong Diameter Padded Decompositions
Arnold Filtser
In Approx 2019.
[ arXiv version ] [ Conference version ] [ Video ] [ Slides ]Given a weighted graph $G=(V,E,w)$, a partition of $V$ is $Δ$-bounded if the diameter of each cluster is bounded by $Δ$. A distribution over $Δ$-bounded partitions is a $β$-padded decomposition if every ball of radius $γΔ$ is contained in a single cluster with probability at least $e^{-β\cdotγ}$. The weak diameter of a cluster $C$ is measured w.r.t. distances in $G$, while the strong diameter is measured w.r.t. distances in the induced graph $G[C]$. The decomposition is weak/strong according to the diameter guarantee.
Formerly, it was proven that $K_r$ minor free graphs admit weak decompositions with padding parameter $O(r)$, while for strong decompositions only $O(r^2)$ padding parameter was known. Furthermore, for the case of a graph $G$, for which the induced shortest path metric $d_G$ has doubling dimension $d$, a weak $O(d)$-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known.
We construct strong $O(r)$-padded decompositions for $K_r$ minor free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension $d$ we construct a strong $O(d)$-padded decomposition, which is also tight. We use this decomposition to construct strong $\left(O(d),\tilde{O}(d)\right)$ sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.
- Constructing Light Spanners Deterministically in Near-Linear Time
Stephen Alstrup, Søren Dahlgaard, Arnold Filtser, Morten Stöckel, Christian Wulff-Nilsen
In ESA 2019.
In TCS 2022 (Theoretical Computer Science).
[ arXiv version ] [ Journal version ] [ Conference version ]Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [CW18] improved the state-of-the-art for light spanners by constructing a $(2k-1)(1+ε)$-spanner with $O(n^{1+1/k})$ edges and $O_ε(n^{1/k})$ lightness. Soon after, Filtser and Solomon [FS19] showed that the classic greedy spanner construction achieves the same bounds The major drawback of the greedy spanner is its running time of $O(mn^{1+1/k})$ (which is faster than [CW16]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness $Ω_ε(kn^{1/k})$, even when randomization is used. The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an $O_ε(n^{2+1/k+ε'})$ time spanner construction which achieves the state-of-the-art bounds. Our second result is an $O_ε(m + n\log n)$ time construction of a spanner with $(2k-1)(1+ε)$ stretch, $O(\log k\cdot n^{1+1/k})$ edges and $O_ε(\log k\cdot n^{1/k})$ lightness. This is an exponential improvement in the dependence on $k$ compared to the previous result with such running time. Finally, for the important special case where $k=\log n$, for every constant $ε>0$, we provide an $O(m+n^{1+ε})$ time construction that produces an $O(\log n)$-spanner with $O(n)$ edges and $O(1)$ lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any $k = ω(1)$. To achieve our constructions, we show a novel deterministic incremental approximate distance oracle, which may be of independent interest.
- Relaxed Voronoi: a Simple Framework for Terminal-Clustering Problems
Arnold Filtser, Robert Krauthgamer, Ohad Trabelsi
In SOSA 2019.
[ arXiv version ] [ Conference version ]We reprove three known algorithmic bounds for terminal-clustering problems, using a single framework that leads to simpler proofs. In this genre of problems, the input is a metric space $(X,d)$ (possibly arising from a graph) and a subset of terminals $K\subset X$, and the goal is to partition the points $X$ such that each part, called a cluster, contains exactly one terminal (possibly with connectivity requirements) so as to minimize some objective. The three bounds we reprove are for Steiner Point Removal on trees [Gupta, SODA 2001], for Metric $0$-Extension in bounded doubling dimension [Lee and Naor, unpublished 2003], and for Connected Metric $0$-Extension [Englert et al., SICOMP 2014].
A natural approach is to cluster each point with its closest terminal, which would partition $X$ into so-called Voronoi cells, but this approach can fail miserably due to its stringent cluster boundaries. A now-standard fix, which we call the Relaxed-Voronoi framework, is to use enlarged Voronoi cells, but to obtain disjoint clusters, the cells are computed greedily according to some order. This method, first proposed by Calinescu, Karloff and Rabani [SICOMP 2004], was employed successfully to provide state-of-the-art results for terminal-clustering problems on general metrics. However, for restricted families of metrics, e.g., trees and doubling metrics, only more complicated, ad-hoc algorithms are known. Our main contribution is to demonstrate that the Relaxed-Voronoi algorithm is applicable to restricted metrics, and actually leads to relatively simple algorithms and analyses.
- Light Spanners for High Dimensional Norms via Stochastic Decompositions
Arnold Filtser, Ofer Neiman
In ESA 2018.
In Algorithmica 2022
[ arXiv version ] [ Journal version ] [ Conference version ] [ Full version ]Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any $n$-point Euclidean metric has an $O(t)$-spanner with $\tilde{O}(n^{1+1/t^2})$ edges, little is known.
In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of $\ell_p$ with $1<p\le 2$. Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any $n$-point subset of $\ell_p$ for $1<p\le 2$ has an $O(t)$-spanner with $n^{1+\tilde{O}(1/t^p)}$ edges and lightness $n^{\tilde{O}(1/t^p)}$.
In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an $O(t)$-spanner with lightness $O(n^{1/t})$. We exhibit the following tradeoff: metrics with decomposability parameter $ν=ν(t)$ admit an $O(t)$-spanner with lightness $\tilde{O}(ν^{1/t})$. For example, $n$-point Euclidean metrics have $ν\le n^{1/t}$, metrics with doubling constant $λ$ have $ν\leλ$, and graphs of genus $g$ have $ν\le g$. While these families do admit a ($1+ε$)-spanner, its lightness depend exponentially on the dimension (resp. $\log g$). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.
- Metric Embedding via Shortest Path Decompositions
Ittai Abraham, Arnold Filtser, Anupam Gupta, Ofer Neiman
In STOC 2018.
In SICOMP 2022. (SIAM Journal on Computing)
[ arXiv version ] [ Journal version ] [ Conference version ] [ Video 1 (STOC presentation)] [ Video 2]We study the problem of embedding shortest-path metrics of weighted graphs into $\ell_p$ spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of Shortest Path Decomposition depth is inductively defined: A (weighed) path graph has shortest path decomposition (SPD) depth $1$. General graph has an SPD of depth $k$ if it contains a shortest path whose deletion leads to a graph, each of whose components has SPD depth at most $k-1$. In this paper we give an $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$-distortion embedding for graphs of SPD depth at most $k$. This result is asymptotically tight for any fixed $p>1$, while for $p=1$ it is tight up to second order terms.
As a corollary of this result, we show that graphs having pathwidth $k$ embed into $\ell_p$ with distortion $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$. For $p=1$, this improves over the best previous bound of Lee and Sidiropoulos that was exponential in $k$; moreover, for other values of $p$ it gives the first embeddings whose distortion is independent of the graph size $n$. Furthermore, we use the fact that planar graphs have SPD depth $O(\log n)$ to give a new proof that any planar graph embeds into $\ell_1$ with distortion $O(\sqrt{\log n})$. Our approach also gives new results for graphs with bounded treewidth, and for graphs excluding a fixed minor.
- Steiner Point Removal with Distortion O(log k)
Arnold Filtser
In SODA 2018.
Steiner Point Removal with distortion O(logk), using the Relaxed-Voronoi algorithm
This is the full version of the paper. The title was changed to emphasize the different algorithm analyzed in this paper.
In SICOMP 2019. (SIAM Journal on Computing)
[ arXiv version ] [ Journal version ] [ Conference version ] [ Video ] [ Slides ]
In the Steiner point removal problem, we are given a weighted graph $G=(V,E)$ and a set of terminals $K\subset V$ of size $k$. The objective is to find a minor $M$ of $G$ with only the terminals as its vertex set, such that distances between the terminals will be preserved up to a small multiplicative distortion. Kamma, Krauthgamer, and Nguyen [SIAM J. Comput., 44 (2015), pp. 975–995] devised a ball-growing algorithm with exponential distributions to show that the distortion is at most $O(\log^5 k)$. Cheung [Proceedings of the 29th Annual ACM/SIAM Symposium on Discrete Algorithms, 2018, pp. 1353–1360] improved the analysis of the same algorithm, bounding the distortion by $O(\log^2 k)$. We devise a novel and simpler algorithm (called the Relaxed-Voronoi algorithm) which incurs distortion $O(\log k)$. This algorithm can be implemented in almost linear time ($O(|E|\log |V|)$).
- Ramsey Spanning Trees and their Applications
Ittai Abraham, Shiri Chechik, Michael Elkin, Arnold Filtser, Ofer Neiman
In SODA 2018.
In TALG 2020 (Transactions on Algorithms).
[ arXiv version ] [ Journal version ] [ Conference version ] [ Video ] [ Slides ]The metric Ramsey problem asks for the largest subset $S$ of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor 2007 devised the so called Ramsey Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems.
In this paper we study the natural extension of the metric Ramsey problem to graphs, and introduce the notion of Ramsey Spanning Trees. We ask for the largest subset $S\subseteq V$ of a given graph $G=(V,E)$, such that there exists a spanning tree of $G$ that has small stretch for $S$. Applied iteratively, this provides a small collection of spanning trees, such that each vertex has a tree providing low stretch paths to all other vertices. The union of these trees serves as a special type of spanner, a tree-padding spanner. We use this spanner to devise the first compact stateless routing scheme with $O(1)$ routing decision time, and labels which are much shorter than in all currently existing schemes.
We first revisit the metric Ramsey problem, and provide a new deterministic construction. We prove that for every $k$, any $n$-point metric space has a subset $S$ of size at least $n^{1-1/k}$ which embeds into an ultrametric with distortion $8k$. This results improves the best previous result of Mendel and Naor that obtained distortion $128k$ and required randomization. In addition, it provides the state-of-the-art deterministic construction of a distance oracle. Building on this result, we prove that for every $k$, any $n$-vertex graph $G=(V,E)$ has a subset $S$ of size at least $n^{1-1/k}$, and a spanning tree of $G$, that has stretch $O(k \log \log n)$ between any point in $S$ and any point in $V$.
- Sparsification of Two-Variable Valued CSPs
Arnold Filtser, Robert Krauthgamer
In SIDMA 2017 (Siam journal on discrete mathematics).
[ arXiv version ] [ Journal version ]A valued constraint satisfaction problem (VCSP) instance $(V,Π,w)$ is a set of variables $V$ with a set of constraints $Π$ weighted by $w$. Given a VCSP instance, we are interested in a re-weighted sub-instance $(V,Π'\subset Π,w')$ such that preserves the value of the given instance (under every assignment to the variables) within factor $1\pmε$. A well-studied special case is cut sparsification in graphs, which has found various applications.
We show that a VCSP instance consisting of a single boolean predicate $P(x,y)$ (e.g., for cut, $P=\mbox{XOR}$) can be sparsified into $O(|V|/ε^2)$ constraints if and only if the number of inputs that satisfy $P$ is anything but one (i.e., $|P^{-1}(1)| \neq 1$). Furthermore, this sparsity bound is tight unless $P$ is a relatively trivial predicate. We conclude that also systems of 2SAT (or 2LIN) constraints can be sparsified.
- Distributed Monitoring of Election Winners
Arnold Filtser, Nimrod Talmon
In AAMAS 2017. Was nominated for best paper award. (what?!)
In AIJ 2019 (Artificial Intelligence Journal).
[ arXiv version ] [ Journal version ] [ Conference version ] [ Full version ]We consider distributed elections, where there is a center and $k$ sites. In such distributed elections, each voter has preferences over some set of candidates, and each voter is assigned to exactly one site such that each site is aware only of the voters assigned to it. The center is able to directly communicate with all sites. We are interested in designing communication-efficient protocols, allowing the center to maintain a candidate which, with arbitrarily high probability, is guaranteed to be a winner, or at least close to being a winner. We consider various single-winner voting rules, such as variants of Approval voting and scoring rules, tournament-based voting rules, and several round-based voting rules. For the voting rules we consider, we show that, using communication which is logarithmic in the number of voters, it is possible for the center to maintain such approximate winners; that is, upon a query at any time the center can immediately return a candidate which is guaranteed to be an approximate winner with high probability. We complement our protocols with lower bounds. Our results are theoretical in nature and relate to various scenarios, such as aggregating customer preferences in online shopping websites or supermarket chains and collecting votes from different polling stations of political elections.
- The Greedy Spanner is Existentially Optimal
Arnold Filtser, Shay Solomon
In PODC 2016. Best student paper.
In SICOMP 2020 (SIAM Journal on Computing).
[ arXiv version ] [ Journal version ] [ Conference version ] [ Full version ] [ Video ]The greedy spanner is arguably the simplest and most well-studied spanner construction. Experimental results demonstrate that it is at least as good as any other spanner construction, in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive.
In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is existentially optimal (or existentially near-optimal) for several important graph families, in terms of both the size and weight. Roughly speaking, the greedy spanner is said to be existentially optimal (or near-optimal) for a graph family $\mathcal G$ if the worst performance of the greedy spanner over all graphs in $\mathcal G$ is just as good (or nearly as good) as the worst performance of an optimal spanner over all graphs in $\mathcal G$.
Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphs (due to Chechik and Wulff-Nilsen [SODA'16]) and doubling metrics (due to Gottlieb [FOCS'15]) are complex. Plugging our observation on these results, we conclude that the greedy spanner achieves near-optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area.
Further, we observe that approximate-greedy spanners are existentially near-optimal as well. Consequently, we provide an $O(n \log n)$-time construction of $(1+ε)$-spanners for doubling metrics with constant lightness and degree. Our construction improves Gottlieb's construction, whose runtime is $O(n \log^2 n)$ and whose number of edges and degree are unbounded, and remarkably, it matches the state-of-the-art Euclidean result (due to Gudmundsson et al. [SICOMP'02]) in all the involved parameters (up to dependencies on $ε$ and the dimension).
- On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion
Yair Bartal, Arnold Filtser, Ofer Neiman
In SODA 2016.
In JCSS 2019. (Journal of Computer and System Sciences)
[ arXiv version ] [ Journal version ] [ Conference version ] [ Video ] [ Slides ]This paper makes two main contributions: The first is the construction of a near-minimum spanning tree with constant average distortion. The second is a general equivalence theorem relating two refined notions of distortion: scaling distortion and prioritized distortion.
Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [ABN15] that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a spanning tree whose weight is at most $(1+ρ)$ times that of the MST, providing constant average distortion $O(1/ρ)$. Our result exhibits the best possible tradeoff of this type.
This result makes use of a general equivalence theorem relating two recently developed notions of distortion for metric embedding. The first is the notion of scaling distortion, which provides improved distortion for $1-ε$ fractions of the pairs, for all $ε$ simultaneously. A stronger version called coarse scaling distortion, has improved distortion guarantees for the furthest pairs. The second notion is that of prioritized distortion, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation.
- Prioritized Metric Structures and Embedding
Michael Elkin, Arnold Filtser, Ofer Neiman
In STOC 2015.
In SICOMP 2018. (SIAM Journal on Computing)
[ arXiv version ] [ Journal version ] [ Conference version ] [ Full version ]Metric data structures (distance oracles, distance labeling schemes, routing schemes) and low-distortion embeddings provide a powerful algorithmic methodology, which has been successfully applied for approximation algorithms [llr], online algorithms [BBMN11], distributed algorithms [KKMPT12] and for computing sparsifiers [ST04]. However, this methodology appears to have a limitation: the worst-case performance inherently depends on the cardinality of the metric, and one could not specify in advance which vertices/points should enjoy a better service (i.e., stretch/distortion, label size/dimension) than that given by the worst-case guarantee.
In this paper we alleviate this limitation by devising a suit of prioritized metric data structures and embeddings. We show that given a priority ranking $(x_1,x_2,\ldots,x_n)$ of the graph vertices (respectively, metric points) one can devise a metric data structure (respectively, embedding) in which the stretch (resp., distortion) incurred by any pair containing a vertex $x_j$ will depend on the rank $j$ of the vertex. We also show that other important parameters, such as the label size and (in some sense) the dimension, may depend only on $j$. In some of our metric data structures (resp., embeddings) we achieve both prioritized stretch (resp., distortion) and label size (resp., dimension) simultaneously. The worst-case performance of our metric data structures and embeddings is typically asymptotically no worse than of their non-prioritized counterparts.
- Terminal Embeddings
Michael Elkin, Arnold Filtser, Ofer Neiman
In Approx 2015.
In TCS 2017 (Theoretical Computer Science).
[ arXiv version ] [ Journal version ] [ Conference version ]In this paper we study terminal embeddings, in which one is given a finite metric $(X,d_X)$ (or a graph $G=(V,E)$) and a subset $K \subseteq X$ of its points are designated as terminals. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve $\approx|K|\cdot |X|$ pairs, the distortion depends only on $|K|$, rather than on $|X|$.
We also strengthen this notion, and consider embeddings that approximately preserve the distances between all pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to $X \times X$ and with respect to $K \times X$.
Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, [ALN08] devised an $\tilde{O}(\sqrt{\log r})$-approximation algorithm for sparsest-cut instances with $r$ demands. Building on their framework, we provide an $\tilde{O}(\sqrt{\log |K|})$-approximation for sparsest-cut instances in which each demand is incident on one of the vertices of $K$ (aka, terminals). Since $|K| \le r$, our bound generalizes that of [ALN08].
- Efficient determination of the unique decodability of a string
Arnold Filtser, Jiaxi Jin, Aryeh Kontorovich, Ari Trachtenberg
In ISIT 2013.
[ Conference version ]Determining whether an unordered collection of overlapping substrings (called shingles) can be uniquely decoded into a consistent string is a problem common to a broad assortment of disciplines ranging from networking and information theory through cryptography and even genetic engineering and linguistics. We present a new insight that yields an efficient streaming algorithm for determining whether a string of $n$ characters over the alphabet $\Sigma$ can be uniquely decoded from its two-character shingles; our online algorithm achieves an overall time complexity $\Theta(n+|\Sigma|)$ and space complexity $O(|\Sigma|)$. As a motivating application, we demonstrate how this algorithm can be adapted to larger, varying-size shingles for (empirically) efficient string reconciliation.
Supervisors: Michael Elkin and Ofer Neiman.
Supervisors: Robert Krauthgamer and Ofer Neiman.
Teaching:
- Spring 2019:
- Spring 2018:
- Spring 2017:
- Spring 2016:
- Spring 2015:
- Fall 2015:
- Spring 2014:
- Fall 2014:
- Spring 2013:
- Fall 2013:
Grading:
- Fall 2019:
- Fall 2018:
- Fall 2017:
- Fall 2016:
Team
I am recruiting excellent graduate students to our research group. If you are interested, don't hesitate to shoot me an email!
Current and past members:
MSc Students.
- Gal Ben-Ami (2025-now)
- Omer Badichi (2025-now)
Postdocs.
- Shinwoo An (2025-now)
- Orr Fischer (2024-now)
- Ameet Gadekar (2023-2024)
Best Relative Award
The Best Relative Award is given annually by the Filtser family cooperation*, to a family member of the Filtser family, as recognition for great investment and donation to the family.
Best Relative Award Laureates:
- 2015 - Emi Filtser - for successful graduating from high school.
- 2014 - Yosi Filtser - for successful discharging from the military (and staying alive).
- 2013 - Naama Filtser - for being born.
- 2012 - Omrit Filtser - for creating an infrastructure for the family to spread and multiply.
*The head and sole member of the Filtser family cooperation is Arnold Filster.
Personal
I am married to Omrit Filtser (who is also in theory!), and father of Naama, Hadass, Ehud Emmanuel, and Boaz Lazer.
